The Binomial Experiment and the Binomial Formula (6.5)
Summary
TLDRThis video script offers an insightful exploration of the binomial setting and formula within the context of probability distribution. It explains the concept of binomial experiments, which involve fixed trials with two outcomes, constant success probability, and independent trials. Through examples like coin flipping and marble drawing, the script demonstrates how to calculate probabilities of specific outcomes and verifies binomial conditions. It also introduces the binomial formula as a shortcut for these calculations, emphasizing its applicability to binomial experiments only.
Takeaways
- 📚 The binomial setting and formula are discussed, focusing on the probability of success or failure in repeated experiments.
- 🔢 The prefix 'bi' signifies two outcomes, such as success or failure, in binomial probabilities.
- 🧩 Four conditions must be met for a binomial setting: fixed number of trials, two outcomes per trial, constant probability of success, and independence of trials.
- 🪙 An example of a binomial experiment is flipping a coin multiple times, calculating the probability of getting a certain number of heads.
- 🎲 The probability of getting exactly one head in three coin flips is calculated by considering all possible outcomes and their probabilities.
- 📉 Each outcome's probability is calculated using the product of the probabilities of individual flips, which in this case is 0.125.
- 📈 The total probability of getting exactly one head is the sum of the probabilities of all individual outcomes, resulting in 0.375.
- 🟢 The script checks if the coin flip experiment is binomial by confirming it satisfies all four binomial conditions.
- 🔄 Another example involves drawing marbles with replacement from a box, illustrating the binomial setting with a different context.
- 🎯 The probability of drawing exactly two green marbles from a box of ten is calculated using both direct enumeration and the binomial formula.
- 📝 The binomial formula provides a shortcut for calculating probabilities in binomial experiments, represented as 'n choose k' times the success probability raised to the power of successes, times the failure probability raised to the power of failures.
Q & A
What is the binomial probability distribution?
-The binomial probability distribution refers to the probability of a success or failure in an experiment that is repeated multiple times, resulting in two possible outcomes for each trial.
What does the prefix 'bi' signify in the term 'binomial'?
-The prefix 'bi' signifies two, as seen in words like 'bicycle' and 'binoculars', and in the context of binomial probabilities, it refers to the two outcomes: a success or a failure.
What are the four conditions that must be satisfied for a setting to be considered binomial?
-The four conditions are: 1) a fixed number of trials, 2) only two possible outcomes for each trial, 3) the probability of success must be constant for every trial, and 4) each trial must be independent of the others.
Can you provide an example of a binomial experiment?
-An example of a binomial experiment is flipping a regular coin three times, where the probability of getting heads (success) or tails (failure) remains constant and each flip is independent.
How many possible ways are there to get exactly one head when flipping a coin three times?
-There are three possible ways to get exactly one head when flipping a coin three times: H-T-T, T-H-T, and T-T-H.
What is the probability of getting exactly one head when flipping a coin three times?
-The probability of getting exactly one head when flipping a coin three times is 0.375, which is calculated by adding the probabilities of the three different outcomes (0.125 each).
What does it mean for the trials in a binomial experiment to be independent?
-For trials to be independent in a binomial experiment means that the outcome of one trial does not influence the outcome of another trial.
How does the binomial formula help in calculating probabilities in a binomial experiment?
-The binomial formula provides a shortcut for calculating probabilities by using the number of trials, the number of successes, and the probability of success, without having to list all possible outcomes.
What is the probability of drawing exactly two green marbles from a box of 10 marbles with 3 pink, 2 green, and 5 blue marbles when drawing 5 marbles with replacement?
-The probability is 0.2048, calculated using the binomial formula or by adding the probabilities of all possible outcomes where exactly two green marbles are drawn.
What is the binomial formula and how is it structured?
-The binomial formula is structured as P(k) = n choose k * (p^k) * ((1-p)^(n-k)), where k is the number of successes, n is the number of trials, p is the probability of success, and 'n choose k' is the combination formula representing the number of ways to choose k successes from n trials.
Why is it important to check if all four conditions are satisfied before applying the binomial formula?
-It is important to ensure that all four conditions are satisfied to confirm that the experiment is a binomial experiment, as the binomial formula can only be applied to binomial settings.
Outlines
📚 Introduction to Binomial Setting and Formula
This paragraph introduces the concept of the binomial setting and formula in the context of probability. It explains that the binomial probability distribution is about the likelihood of success or failure in repeated experiments. The prefix 'bi' is highlighted to emphasize the two possible outcomes. The binomial setting is defined by four conditions: a fixed number of trials, two possible outcomes per trial, constant probability of success, and independence of trials. An example of flipping a coin three times is used to illustrate these concepts, with a step-by-step explanation of how to calculate the probability of getting exactly one head. The paragraph concludes by confirming that the coin flip scenario is a binomial experiment due to the satisfaction of all four conditions.
🎲 Applying the Binomial Setting to a Marble Drawing Experiment
The second paragraph delves into a more complex example involving drawing marbles from a box with replacement. It first establishes whether the scenario meets the binomial setting criteria, confirming that it does due to a fixed number of trials, two outcomes (success or failure), constant probability of success, and independent trials. The probability of drawing a green marble is calculated as 0.2, and the probability of not drawing a green marble as 0.8. The paragraph then explores the different ways of drawing exactly two green marbles out of five trials, emphasizing the uniform probability of 0.2048 for each of the ten possible outcomes. It concludes by demonstrating the use of the binomial formula as a shortcut for calculating the probability of exactly two successes in a binomial experiment, yielding the same result as the manual calculation.
Mindmap
Keywords
💡Binomial Setting
💡Binomial Probability Distribution
💡Success and Failure
💡Binomial Experiment
💡Probability of Success
💡Independence
💡Combination Formula
💡Binomial Formula
💡Trials
💡Probability Calculation
💡Supporting the Video
Highlights
Introduction to the binomial setting and formula in probability distribution.
Explanation of the binomial probability distribution involving success or failure in repeated experiments.
The prefix 'bi' signifies two outcomes: success or failure in binomial probabilities.
Four conditions required for the binomial setting: fixed number of trials, two outcomes, constant probability of success, and independence of trials.
Example of a binomial experiment: flipping a coin three times to find the probability of getting exactly one head.
Calculating probabilities by considering different outcomes and their respective probabilities.
Verification of a binomial experiment by checking if it satisfies the four binomial conditions.
Demonstration of calculating the probability of drawing exactly two green marbles from a box of 10 marbles with replacement.
Use of the combination formula in scientific calculators to simplify binomial probability calculations.
Explanation of how to determine the probability of a success and a failure in a binomial setting.
Calculation of the probability of drawing exactly two green marbles using both direct calculation and the binomial formula.
The binomial formula provides a shortcut for calculating probabilities in binomial experiments.
Emphasizing the importance of ensuring a binomial setting before applying the binomial formula.
Practical application of the binomial formula to solve the marble drawing problem.
Final calculation and verification of the probability using the binomial formula.
Encouragement to support the creators for more educational content on Patreon and the website.
Transcripts
in this video we'll be learning about
the binomial setting and the binomial
formula when we talk about the binomial
probability distribution we are
referring to the probability of a
success or a failure in an experiment
that is repeated multiple times you can
easily remember this by paying attention
to the prefix bi which literally means
two for example in bicycles there are
two wheels and in binoculars there are
two lenses
however in binomial probabilities there
are two outcomes a success or a failure
before we do some practice problems we
need to talk about the binomial setting
the binomial setting must satisfy four
conditions the first condition is that
the number of trials n must be fixed the
second condition is that there are only
two possible outcomes for each trial you
can either have a success or a failure
the third condition is that the
probability of success must be constant
for every trial and finally the fourth
condition is that each trial must be
independent this means that the outcome
of one trial does not influence the
outcome of another trial an experiment
that satisfies these four conditions is
called a binomial experiment the
binomial setting will make more sense
after we do an example so let's jump
into it if you flip a regular coin three
times what is the probability of getting
exactly one head and is this a binomial
experiment feel free to pause the video
at this point so you can try this
question for yourself
since the question says that we are
flipping the coin three times I will
have three blank spaces one for each
flip the first blank space is for the
first flip the second is for the second
flip and the third is for the third flip
in order to solve this problem we have
to realize that there are only three
possible ways for us to get exactly one
head the first way is to get heads on
the first flip and then tails on the
second and third flip the second way is
to get tails on the first flip heads on
the second flip and then tails on the
third flip the third and final way is to
get tails on the first and second flip
and then getting heads on the third flip
so overall we see that there are three
different outcomes where we get exactly
one head and we see that they vary based
on the order now all we have to do is
calculate the probabilities of each
outcome and add them together to get the
answer we know that the probability for
getting heads is 0.5 and the probability
of getting tails is also 0.5 to
calculate the probability of the first
outcome we will multiply the probability
of heads times the probability of tails
times the probability of tails again
this is equal to 0.5 times 0.5 times 0.5
and this is equal to 0.125 when we
calculate the probabilities for the
second and third outcome you'll find
that they will also be equal to 0.125
now all we have to do is add these
probabilities together to get the answer
and when we do we get the probability of
getting exactly one head which is equal
to 0.375
to check if this is a binomial
experiment we have to see if it
satisfies the for binomial conditions
the first condition is satisfied because
we have a fixed number of trials n is
equal to 3 because the experiment has to
be repeated 3 times in other words the
coin has to be flipped three times the
second condition is also satisfied
because we can define a success as
getting heads and we can define a
failure as not getting heads this is the
same as saying getting tails is a
failure
the third condition is also satisfied
because the probability of success
remained constant for every trial in
other words the probability of getting
heads is 0.5 and it stayed that way
every time the coin was flipped and
finally the fourth condition is also
satisfied because each trial is
independent gaining heads or tails for
one trial doesn't change the probability
of getting heads or tails for the other
trials since all four conditions are
satisfied we know that this is a
binomial experiment now let's do a
harder problem suppose we have 10
marbles in a box we have three pink
marbles two green marbles and five blue
marbles if we pick all five marbles with
replacement what is the probability of
drawing exactly two green marbles and is
this a binomial experiment feel free to
pause the video at this point so you can
try this question for yourself
before we calculate any probabilities
let's see if we have a binomial setting
is there a fixed number of trials yes n
is equal to 5 because we are picking out
5 marbles are the two possible outcomes
a success and a failure yes a success is
getting a green marble and a failure is
not getting a green marble
is the probability of success constant
for each trial yes because we are doing
the experiment with replacement every
time we conduct a trial or randomly pick
out one marble we put it back into the
box before drawing another marble if we
didn't do this the trials would become
dependent on one another instead of
being independent in this case the
probability of success is equal to the
probability of getting one green marble
which is equal to two over ten or 0.2
our trials independent of each other for
the reasons mentioned previously the
answer is yes since all four conditions
are satisfied we know that this is the
binomial experiment to solve this
problem we need to write out all the
possible ways of drawing exactly two
green marbles for example one way is
drawing a green marble on the first and
second try and then not drawing a green
marble on the third fourth and fifth try
another way could be only drawing a
green marble on the second and fifth try
notice that I decided to write a dashed
line for not getting a green marble this
is because I don't care if the marble
was blue or pink if I don't get a green
marble it's a failure in a binomial
setting we only care if we got a failure
or a success overall we see that there
are ten different ways of drawing two
green marbles we can properly reframe
this by saying that there are ten
different ways of getting two successes
and three failures now let's calculate
some probabilities the probability of a
success is equal to the probability of
drawing a green marble since there are
ten marbles in the box and only two of
them are green the probability of
drawing a green marble is equal to two
over ten or 0.2 therefore the
probability of a success is equal to 0.2
the probability of a failure is equal to
their probability of not drawing a green
marble there are 10 marbles in the box
and eight of them aren't green so the
probability of not drawing a green
marble is just 8 over 10 or 0.8
therefore the probability of a failure
is equal to 0.8 let's calculate the
probability of the first outcome we have
two successes followed by three failures
so we will have 0.2 times 0.2 times 0.8
times 0.8 times 0.8 which gives us an
answer of zero point zero two zero four
eight if we do a similar calculation for
the rest of the outcomes you should also
get an answer of zero point zero two
zero four eight notice how each of these
ten outcomes has the same probability
the reason for this is because each
outcome has two successes and three
failures so it makes sense that they all
have the same probabilities now to
calculate the probability of drawing
exactly two green marbles all we have to
do is add up all these probabilities
together and when we do we get an answer
of zero point two zero four eight
however there is another way we can
calculate this answer and that is by
using the binomial formula the binomial
formula looks like this it looks a
little scary but let's break it down k
is the number of successes n is the
number of trials and little P is the
probability of success what we have here
in brackets represents the combination
formula it is often seen in scientific
calculators as NCR which is the same
thing as saying n choose R so for the
binomial formula we say that the
probability of k is equal to n choose k
times the probability of success little
P raised to the number of successes K
times the probability of failure which
is 1 minus P raised to the number of
failures which is equal to n minus K
by using this formula we are essentially
giving ourselves a nice shortcut for
calculating things however it's
important to remember that we can only
use this formula if we have a binomial
experiment let's go back to the problem
so I can show you how to use this
formula if we pickle five marbles with
replacement what is the probability of
drawing exactly two green marbles in
this problem and is equal to five since
there are five trials in other words we
are randomly picking out five marbles K
is equal to two since we are only
concerned about getting exactly two
successes this is the same as getting
exactly two green marbles little P is
equal to zero point two as this is the
probability of success that we
calculated earlier now all we have to do
is plug these values into the formula
and we'll get an answer so the
probability of drawing exactly two green
marbles in other words the probability
of getting two successes is equal to 5
choose 2 times 0.2 squared times 1 minus
0.2 raised to the power of 5 minus 2
after this calculation we get an answer
of zero point two zero four eight which
is the same answer that we calculated
from before
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